Phonon-Induced Decay of the Electron Spin in Quantum Dots

We study spin relaxation and decoherence in a GaAs quantum dot due to spin-orbit (SO) interaction. We derive an effective Hamiltonian which couples the electron spin to phonons or any other fluctuation of the dot potential. We show that the spin decoherence time $T_2$ is as large as the spin relaxation time $T_1$, under realistic conditions. For the Dresselhaus and Rashba SO couplings, we find that, in leading order, the effective $B$ field can have only fluctuations transverse to the applied $B$ field. As a result, $T_2=2T_1$ for arbitrarily large Zeeman splittings, in contrast to the naively expected case $T_2\ll T_1$. We show that the spin decay is drastically suppressed for certain $B$-field directions and ratios of SO coupling constants. Finally, for the spin-phonon coupling, we show that $T_2=2T_1$ for all SO mechanisms in leading order in the electron-phonon interaction.

Figure 1

Solid curve: The relaxation rate $1/T_1$ of Eq. (18) as a function of an in-plane $B$ for a GaAs QD with $\mathrm{\hbar }{\omega }_0=1.1\mathrm{m}\mathrm{e}\mathrm{V}$, $d=5\mathrm{n}\mathrm{m}$, ${\lambda }_{\mathrm{S}\mathrm{O}}=\mathrm{\hbar }/m^{*}\beta =1\mu \mathrm{m}$, and $\alpha =0$. Dashed (dotted) curve: Contribution of the piezoelectric mechanism (${\overline{\beta }}_{j\vartheta }$) with transverse (longitudinal) phonons. Dot-dashed curve: Contribution of the deformation potential mechanism (${\overline{\Xi }}_j$). Note that $1/T_1\propto 1/{\lambda }_{\mathrm{S}\mathrm{O}}^2$.